Understanding $E(1_{[X>n]}) = P[X>n]$

Question

I'm checking some notes in measure and integration theory, and found this statement:

$E(1_{[X>n]}) = P[X>n]$

Just need to understand it.

Is this the correct interpretation? Please need some advise.

I think $E(1_{[X>n]}) = \int_{X>n} dP $.

But I'm not able to link this integral to $P[X>n]$.

Thanks!

Answer 1

If you write out the expectation:

$$\int 1_{X>n} dP_X = \int_{X\leq n} 1_{X>n} dP_X + \int_{X>n} 1_{X>n} dP_X = \int_{X\leq n} 0\; dP_X + \int_{X>n} 1 dP_X = P_X((n,\infty))$$

You see that you are only integrating over the set where $X>n$, the rest of the domain of the probability measure is zeroed out by the indicator function.